\[ \int \frac {\sqrt {c+d \tan (e+f x)}}{a+b \tan (e+f x)} \, dx \]
Optimal antiderivative \[ \frac {\arctanh \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {-i d +c}}\right ) \sqrt {-i d +c}}{\left (i a +b \right ) f}-\frac {\arctanh \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {i d +c}}\right ) \sqrt {i d +c}}{\left (i a -b \right ) f}-\frac {2 \arctanh \left (\frac {\sqrt {b}\, \sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {-a d +b c}}\right ) \sqrt {b}\, \sqrt {-a d +b c}}{\left (a^{2}+b^{2}\right ) f} \]
command
integrate((c+d*tan(f*x+e))^(1/2)/(a+b*tan(f*x+e)),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \text {output too large to display} \]
Fricas 1.3.7 via sagemath 9.3 output \[ \text {Timed out} \]